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Toric extensions of Pólya's theorem

Lorenzo Baldi, Rainer Sinn, Máté L. Telek, Julian Weigert

TL;DR

The article extends Pólya's classical certificate for strict copositivity to sparse polynomials by embedding them in toric geometry and using both standard and Cox-coordinate approaches. It introduces the Representation Theorem with Archimedean preprimes to obtain multiplier certificates, and provides explicit certificates when conv$(A)$ is a product of simplices. By connecting sparse copositivity to toric varieties via $X_A$ and $Y_{\\hat A}$, it both generalizes Pólya-type certificates and explores limitations outside structured polytopes. The work further applies these certificates to Symanzik polynomials, showing how copositivity criteria guarantee convergence of Feynman integrals and clarifying the role of the Euclidean region in physically relevant diagrams.

Abstract

The classical version of Pólya's theorem provides a simple method for certifying that a homogeneous polynomial of degree d is strictly copositive, that is, it takes only positive values on the nonnegative real orthant. However, this method might fail to detect copositivity of polynomials that are missing certain degree d monomials. In this paper, we present extensions and converses to Pólya's theorem for sparse polynomials, using techniques from positive toric geometry. Furthermore, we explore how this method can be used to study the convergence of Feynman integrals in particle physics.

Toric extensions of Pólya's theorem

TL;DR

The article extends Pólya's classical certificate for strict copositivity to sparse polynomials by embedding them in toric geometry and using both standard and Cox-coordinate approaches. It introduces the Representation Theorem with Archimedean preprimes to obtain multiplier certificates, and provides explicit certificates when conv is a product of simplices. By connecting sparse copositivity to toric varieties via and , it both generalizes Pólya-type certificates and explores limitations outside structured polytopes. The work further applies these certificates to Symanzik polynomials, showing how copositivity criteria guarantee convergence of Feynman integrals and clarifying the role of the Euclidean region in physically relevant diagrams.

Abstract

The classical version of Pólya's theorem provides a simple method for certifying that a homogeneous polynomial of degree d is strictly copositive, that is, it takes only positive values on the nonnegative real orthant. However, this method might fail to detect copositivity of polynomials that are missing certain degree d monomials. In this paper, we present extensions and converses to Pólya's theorem for sparse polynomials, using techniques from positive toric geometry. Furthermore, we explore how this method can be used to study the convergence of Feynman integrals in particle physics.

Paper Structure

This paper contains 17 sections, 25 theorems, 110 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Applications of copositivity.
  3. Organization.
  4. Notation.
  5. Acknowledgments.
  6. Sparse copositivity
  7. Sparse copositive polynomials
  8. Positive toric geometry
  9. Pólya's theorem for sparse polynomials
  10. The Representation Theorem
  11. First extension of Pólya's theorem
  12. Pólya's theorem via Cox coordinates
  13. Cox homogenization
  14. Preprimes in the Cox ring
  15. Product of simplices
  16. ...and 2 more sections

Key Result

Theorem A

Let $A \subseteq \mathbb{Z}^n$ be a finite set and $k \in \mathbb{N}$. Let $f$ be a (Laurent) polynomial whose support is contained in $k \cdot A$. Then $f$ is strictly $A$-copositive if and only if there exists $N \in \mathbb{N}$ such that $(\sum_{a \in A}t^a)^N f$ has nonnegative coefficients and

Figures (3)

  • Figure 1: Newton polytope of the polynomial $f$ from \ref{['Eq:Running']} sitting inside the $3$-dilated simplex.
  • Figure 2: Newton polytopes of the polynomials considered in Example \ref{['ex:FirstCounterExample']}, \ref{['ex:counterex2']} and \ref{['Ex:MinkowskiSum']}.
  • Figure 3: (a) Newton polytope of the second Symanzik polynomial $\mathcal{F}$ of the banana diagram \ref{['Eq:BananaF']}, $s = -(k_{13} + k_{14} +k_{23} + k_{24})$. (b) depicts the Newton polytope of $\mathcal{F}$ after setting $m_3 = 0$.

Theorems & Definitions (59)

  • Theorem A: see Theorem \ref{['Thm:SparsePolya']}
  • Theorem B: see Theorem \ref{['prop:polyaInCox_standard']}
  • Theorem 2.1: Pólya's theorem Polya
  • Lemma 2.2
  • proof
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • ...and 49 more